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Rev. Roum. Sci. Techn. lectrotechn. et nerg., 53, 4, p. 367376, Bucarest, 2008

ON THE TRANSIENT BEHAVIOR OF SATURATED SYNCHRONOUS MACHINES

AUREL CMPEANU , IOAN CAUIL

Key words : Synchronous machine, Saturation, Modeling and simulation.

In the design stage of synchronous machine is often necessary the predetermination of dynamic characteristics using modeling and simulation. The simulation results and their correlation with reality depend essentially on the way the synchronous machine characterized by high electric and magnetic stress is modeled. Significant data are presented in the following for comparative analyses of different ways to consider the saturation in synchronous machine in dynamic regime.

1. INTRODUCTION

The dynamic regime problems and the stability of the synchronous machine (actually the main source for obtaining electromagnetic energy) have generated numerous long term and wide researches [79]. Generally, the high precision evaluation of the behavior of the alternative current machines during dynamic processes (start, self-excitation) characterized by large variations of magnetic and electric stresses is closely determined by a correct mathematical modeling. The complicated couplings between the windings generate non linear equations systems, which can be solved numerically.

The mathematical models obtained by using representative phasors under controlled hypotheses, represent a coherent and efficient research base.

In [3] the general mathematical models of the saturated synchronous machine with salient poles for various combinations of the state variables are deduced, considering the cross-coupling inductance.

In the following, using [3], we present and interpret the results of significant simulations which allow the predetermination of the behavior of the synchronous machine, in a given dynamic process.

368 Aurel Cmpeanu, Ioan Cauil 2

2. GENERAL EQUATIONS OF THE SATURATED SYNCHRONOUS MACHINE

The mathematical model of the synchronous machine saturated on the main flux way is considered to have the form:

Td

; 0 0 .d

dqdq dq dq d q Eu u ut= + =XU A BX U (1)

Xdq corresponds to diverse combinations of state variables and A, B are matrices whose form is determined by the choice of the state variables (currents only, flux linkages only or some mixture of currents and flux linkages).

In the following we consider as state variables the currents of the machine windings. In this case, to the state vector

T

dq d q E D Qi i i i i=X correspond the matrices (see [3])

( )( )

2 2

2 2

2 2

2 2

2 2

2

,

0 0

00 0 0 00 0 0 00 0 0 0

s dd dq dd dd dq

dq s qq dq dq qq

dd dq E dd DE dd dq

dd dq DE dd D dd dq

dq qq dq dq Q qq

s s m m

s m s m m

E

D

Q

L L m L L L m LL L m L L L m LL m L L L L L m LL m L L L L L m LL m L L L L m L

R L m L L

L L R L LR

RR

+ + + += + + +

+ +

=

A

B ,

where mdmq LLm /2 = is the saliency coefficient; the computational inductances are

.sin;cos,cossin

,cossin)(,sincos

'

'

'22

22

m

mq

mq

mdmmtqq

mmtdqmmtdd

ii

iiLLL

LLLLLL

==+=

=+= (2)

At equation (1) we add the movement equation

,dd

dd

2

2

tpJ

tpJMm r

== (3)

3 Transient behavior of saturated synchronous machines 369

where

( ) ( )[ ].123 22

qddQqDEm iimiimiiipLm ++=

In the referential system d, q we consider

),sin(2),cos(2 11 == tUutUu qd where

.d 0+= t The speed of the main magnetic field m during the dynamic processes will be d ,

d t = + (4)

where

.sind

dcosd

d1sind

dcosd

d1dd '

'

=

=

t

it

iittt

mdmq

m

mdmq

m

The above notations are those of [3] and used by some authors. Remark: When saturation is taken into account in the conventional way, the

cross-coupling inductance vanishes ( )0=dqL . The classical approach form of the mathematical model is obtained by introducing ( ).'mmmmt iLLL == .

For the simplified computation (with constLL mtm == ) the matrix A becomes invariant and the solving of the system describing the dynamical process is much simpler. In this case computational inductances become

mqqdd LLL == and 0=dqL

3. SIMULATION RESULTS

We study in the following the effects of dynamic saturation ( 0dqL ) and saliency on the behavior of synchronous machine with star winding and nominal voltage Un=220V (see Appendix) at the dynamic asynchronous start and synchronization considering the above theory.

The curves m(), (t), (t) were obtained considering various levels of magnetic saturation (different U) and resistive torques Mr. The proposed model, the


simplified model and the classical model are used for comparison. The simulations were performed in Matlab using a fifth order Runge-Kutta with variable step.

With small supplying voltage (U = 100 V) when the saturation is not present and for the same resistive torque (M r= 0), the three computing methods lead to practically identical results. This was to be expected and validates the general system of equations (1).

For an average saturation level (U = Un = 220 V) some small differences of waveform and amplitude are to be remarked. Repeated simulations showed that synchronization is possible till Mr 2 Nm. In the permanent regime the results are practically identical, which confirm that the three mathematical models are correct. This results were those expected and are not presented.

To underline the necessity of consideration of dynamic saturation in formulation of the mathematical models, in the sequel the characteristics for U = 300 V and different values of the resistive torques rM given by the three models are shown.

Fig. 1a Characteristic (t), (t)

for U = 300 V, Mr = 8 Nm. Fig. 2a Characteristic (t), (t)

for U = 300 V, Mr = 7.5 Nm.

Fig. 1b Characteristic m() for U = 300 V,

Mr = 8 Nm. Fig. 2b Characteristic (t), (t)

for U = 300 V, Mr = 7.5 Nm (detail).


Fig. 2c Characteristic m() for U = 300 V, Mr = 7.5 Nm.

Fig. 2f Characteristic Lmt (t) for U = 300 V, Mr = 7.5 Nm.

Fig. 2d Characteristic m() for U = 300 V,

Mr = 7.5 Nm (detail). Fig. 3a Characteristic m() for U = 300 V,

Mr = 4 Nm.

Fig. 2e Characteristic Lm(t) for U = 300 V,

Mr = 7.5 Nm. Fig. 3b Characteristic m() for U = 300 V,

Mr = 4 Nm (detail).


Fig. 3c Characteristic Lm(t) for U = 300 V,

Mr = 4 Nm. Fig. 4b Characteristic m() for U = 300 V,

Mr = 7.5 Nm, classical model.

Fig. 3d Characteristic Lmt (t) for U = 300 V, Mr = 4 Nm.

Fig. 5a Characteristic (t), (t) for U = 300 V, Mr = 7.5 Nm, simplified model

(Lm = 0,4 H).

Fig. 4a Characteristic (t), (t) for U = 300 V,

Mr = 7.5 Nm, classical model. Fig. 5b Characteristic m() for U = 300 V, Mr = 7.5, Nm simplified model (Lm= 0,4 H).


Fig. 6 Characteristic m() for U = 300 V,Mr = 4 Nm, classical model, (detail).

Fig. 7 Characteristic (t), (t) for U = 300V, Mr=4Nm, simplified model (Lm=0,4H), (detail).

Figs 1, 2, 3 show the characteristics for Mr = 8 Nm, 7.5 Nm and 4 Nm using the proposed model. They suggest an asynchronous operation with a large slip, one practically synchronous operation, and respectively a synchronous operation. Fig. 2a offers interesting information regarding the characteristic (t): the rotating field speed grows from 1n/2 to 1n in a time comparable with the duration of the mechanical transient regime. A reversal of the direction may intervene (in our analyses for U = 100 V, 220 V) and almost always over synchronic oscillations exist, finally damped. This observation is a general characteristic for alternating current machines. One can observe the close connection between the mechanical and electromagnetic transient processes, in fact one electro-mechanic process. One can observe the effect of the Gorges phenomenon on the characteristic ( )t at approximate 1n/2. Finally, the synchronous operation is not obtained. Fig. 2b details the asynchronous sustained operation, where the oscillations of (t) are closely correlated with those of (t).

For rM = 8 Nm (Fig. 1 a) the operation is far from synchronous; the permanent oscillations are present in the curves (t) and (t) with a clear underlining of the Gorges phenomenon which does not allow synchronization.

The characteristics of the Figs. 1b and 2c present the correspondent dynamical torques, also affected by the Gorges phenomenon which finally leads to an asynchronous oscillating operation, and respectively practically synchronous. Fig. 2c also shows the large oscillations of the asynchronous torque in the vicinity of the synchronous speed, and in the detailed Fig. 2d the closed limit cycle represented in bold gives the limits in which m and oscillate permanently. When using the proposed method we take into account mL and mtL (Figs. 2e, 2f) which also oscillate in large limits. Fig. 2 e underlines the effect of sustained asynchronous oscillations on mL .


Fig. 3 presents the same dynamic characteristics as above for Mr = 4 Nm, for which the synchronization is achieved. The detailed representation from Fig. 3b different of Fig. 2d, confirms machine synchronization through oscillations that damp in the well defined point of synchronism. Also, taking into account that synchronization is finally achieved, the damped oscillations of mL and mtL disappear and are replaced by constant values corresponding to the level of magnetic stress of the machine (Figs 3c and 3d).

Many other simulations show that for the high saturation level considered (U = 300 V), the synchronous operation is achieved for Mr < 5 Nm and asynchronous one (with a little bigger than 1n/2) for Mr > 7.8 Nm. In the interval (5-7.8) Nm a practically synchronous operation is achieved.

Finally the results of classical and simplified methods are presented. The Figs 4a and 4b show the same characteristics when the classical model is used for Mr = 7.5 Nm. One can see the previous results are not confirmed. We get an asynchronous operation with a large slip, torque oscillations and repeated reversals of (t). The classic model gives also inconclusive results for values sensibly smaller than Mr = 7.5 Nm.

The method with constant parameters offers results comparable with the proposed method if one considers for constmL = , the final saturated value of the given dynamic regime (in our case H 15.0mL ) ; if we take constmL = corresponding to the case of no saturation ( HLm 4.0 ), the constant mathematical model fails (Figs 5a, 5b).

For Mr = 4 Nm the classical method suggest a practically synchronous operation (compare Fig. 6 with Fig 3b). As above, the simplified method offers results comparable with the proposed method if one considers the saturated value of constmL = (in our case Lm = 0.12 H). For Lm 0.4 H we obtain a practically synchronous operation (Fig. 7) with the same comment as for Figure 2 b.

For Mr = 8 Nm the results and interpretations corresponding to classical and simplified methods are as above and similar to Fig. 4Fig. 7.

4. CONCLUSIONS

Many simulations allow the following general conclusions. For usual voltages (U Un) all three models offer close results and can be

equally used. For high voltages the effect of mtL (of dqL ) becomes important. One can

obtain different characteristics when using the three models. Close characteristics are obtained by the enhanced model and the simplified model for constmL = (if we


take mL of the end of the dynamic regime given by the proposed model). This is an indirect method for the validation of the proposed model. The classic method offer inconclusive results.

The enhanced model is recommended for the analysis of dynamic regimes with high solicitations, electric and magnetic. It allows valuable predeterminations for convenient design of a synchronous machine due to operation into a given dynamical regime.

The classic method is the most inexact. Even the method with constmL = that assumes a small volume of calculations is preferable.

The equation (4) contains valuable information about the velocity (t) of the main rotating magnetic field during the transient processes. A close connection between (t) and (t) is observed in all simulations shown above.

Finally we insist on the computational character of the inductances dqqqdd LLL ,, which depend on saturation, reference frame and state variables.

APPENDIX

The motor parameters used for simulation are Rs = 2 , Ls = 0.017 H, LE = 0.0221 H, LD = 0.0204 H,

LQ = 0.0187 H, LDE = 0.00884 H, RE = 0.6 , RD = 0.6 , RQ = 6 , p = 2, J = 0.024 kgm2, m2= 0.6

The characteristics m(im), Lm(im), Lmt(im) are given in Fig. 8.

Fig. 8 Characteristics m(im), Lm(im), Lmt(im).

Received on 17 February, 2008


REFERENCES

1. K. P. Brown, P. Kovacs, P. Vas, A method of including the effects of main flux path saturation in the generalized equations of a. c. machines, IEEE Transactions on PAS-102,1, 1983, pp. 96103.

2. A. N. El-Serafi, J. Wu, Saturation representation in synchronous machine models, Elec. Machines and Power Systems, 20, pp. 355369 (1992).

3. A. Cmpeanu, Nonliniar dynamical models of saturated salient pole synchronous machine, Rev. Roum. Sci. Techn-Electrotech. et Energ., 47, 3, pp. 307317 (2002).

4. A. Cmpeanu, Nonliniar dynamical models for the saturated induction machine, Rev. Roum. Sci. Techn-Electrotech. et Energ., 46, 1, pp. 8999 (2001).

5. A. Cmpeanu, Introduction in ac electrical machines dynamics (in Romanian), Edit. Academiei Romne, Bucharest, 1998.

6. A. Cmpeanu, Transient performance of the saturated induction machine, Electrical Engineering (A. f. E), 78, 4 , pp. 241247. (1995).

7. I. Iglesias, L. Garcia Tabares, I. Tamaret, AD-Q Model for the Self Commutated Synchronous Machine Considering the Effect of Magnetic Saturation, IEEE Trans. Energy Conv., 7, 4, pp. 768776 (1992).

8. E. Levi, Saturation modeling in D-Q axis models of salient pole synchronous machines, IEEE Trans. On Energy Conversion, 14, 1, pp. 4450 (1999).

9. L. Pierrat, E. Dejaeger, M. S. Garrido, Models unification for the saturated synchronous machines, Proc. Int. Conf. on Evolution and Modern Aspects of Synchronous Machines, Zrich, Switzerland, 1991, pp. 4448.

10. S. A. Tahan, I. Kamwa, A two-factor saturation model for synchronous machines with multiple rotor circuits, IEEE Trans. On Energy Conversion, 10, 4, pp. 609616 (1995).

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